JHEP11(2019)170 Published for SISSA by Springer Received: October 7, 2019 Accepted: November 10, 2019 Published: November 28, 2019 Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities Cesar Fierro Cota, a Albrecht Klemm a,b and Thorsten Schimannek c a Bethe Center for Theoretical Physics, Universit¨ at Bonn, D-53115 Bonn, Germany b Hausdorff Center for Mathematics, Universit¨ at Bonn, D-53115 Bonn, Germany c Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria E-mail: fierro@th.physik.uni-bonn.de, aklemm@th.physik.uni-bonn.de, thorsten.schimannek@univie.ac.at Abstract: We calculate the generating functions of BPS indices using their modular properties in Type II and M-theory compactifications on compact genus one fibered CY 3-folds with singular fibers and additional rational sections or just N -sections, in order to study string dualities in four and five dimensions as well as rigid limits in which gravity decouples. The generating functions are Jacobi-forms of Γ 1 (N ) with the complexified fiber volume as modular parameter. The string coupling λ, or the ǫ ± parameters in the rigid limit, as well as the masses of charged hypermultiplets and non-Abelian gauge bosons are elliptic parameters. To understand this structure, we show that specific auto-equivalences act on the category of topological B-branes on these geometries and generate an action of Γ 1 (N ) on the stringy K¨ahler moduli space. We argue that these actions can always be expressed in terms of the generic Seidel-Thomas twist with respect to the 6-brane together with shifts of the B-field and are thus monodromies. This implies the elliptic transformation law that is satisfied by the generating functions. We use Higgs transitions in F-theory to extend the ansatz for the modular bootstrap to genus one fibrations with N -sections and boundary conditions fix the all genus generating functions for small base degrees completely. This allows us to study in depth a wide range of new, non-perturbative theories, which are Type II theory duals to the CHL Z N orbifolds of the heterotic string on K3 × T 2 . In particular, we compare the BPS degeneracies in the large base limit to the perturbative heterotic one-loop amplitude with R 2 + F 2g−2 + insertions for many new Type II geometries. In the rigid limit we can refine the ansatz and obtain the elliptic genus of superconformal theories in 5d. Keywords: D-branes, F-Theory, String Duality, Topological Strings ArXiv ePrint: 1910.01988 Open Access,c  The Authors. Article funded by SCOAP 3 . https://doi.org/10.1007/JHEP11(2019)170